Autumn and Spring

Unit(s) of Assessment	Weighting Towards Module Mark( %)
Coursework: 1 test in Autumn semester	15
Exam: End of Autumn semester	35
Coursework: 1 test in Spring semester	10
Exam: End of Spring semester	40
Qualifying Condition(s)  A weighted aggregate mark of 40% is required to pass the module.

The aim of this module is first to introduce the basic concepts and techniques of vector and matrix calculations, including some applications to geometry and differential equations, and then to use these to motivate the more abstract theory of Linear Algebra.

At the end of the module the student should: ·         Know the terminology associated with vectors in two and three dimensions. ·         Be able to find equations of lines and planes. ·         Use vector methods to prove simple geometrical results. ·         Be able to manipulate matrices and determinants. ·         Be able to solve systems of simultaneous linear equations by several methods. ·         Know how to find eigenvalues and eigenvectors, and carry out matrix diagonalisation. ·         Be able to solve first order systems of homogeneous linear differential equations. ·         Know and understand the terminology associated with vector spaces ·         Understand and be able to apply the relationship between linear maps and matrices. ·         Appreciate the significance of the eigenvalues and eigenvectors of a linear map. ·         Understand quadratic and bilinearforms, and their representation by matrices. ·         Recognise the equation of a conic, and know how to find its principal axes. ·         Understand the concepts of inner product and orthogonality, for vector spaces over the real numbers, complex numbers and finite fields. ·        Be able to formulate simple proofs of results similar to those covered in the course.

·         Vector algebra; scalar, vector and triple products; equations of straight lines and planes.  ·         Geometrical proof by vector methods (e.g. proofs of concurrency and collinearity). ·         Matrices and simultaneous linear equations. Solution by Gaussian elimination. ·         Determinants and invertibility. Eigenvalues and eigenvectors. Matrix diagonalisation.  ·         Functions of matrices, including the matrix exponential. ·         Introduction to vector spaces. Subspaces, linear independence, basis, dimension. ·         Linear transformations. Matrix representation. Kernel and image, rank and nullity. Eigenvectors of a linear map. ·         Systems of first order linear ordinary differential equations (homogeneous, constant coefficients). ·         Quadratic forms. Orthogonal diagonalisation. Geometry of conics, principal axes. ·         Bilinear forms and Inner products. Norms, orthogonality, the Gram-Schmidt process. ·         Complex vector spaces. Unitary and Hermitian matrices. ·         Introduction to finite fields and vector spaces over such fields.

Teaching is by lectures, tutorials and tests: 4 hours per week in each semester.   Learning takes place through lectures, tutorials, tests and exercise sheets.

Recommended:

David Poole, Linear Algebra – A Modern Introduction, Brooks Cole (2002), ISBN 0534341748.

J. Gilbert and C. Jordan (2002), Guide to Mathematical Methods, Palgrave Macmillan, ISBN 0333794443 

Also useful:

S. Goode and S. Annin , Differential Equations and Linear Algebra, Pearson, ISBN 0131293397

C. McGregor, J. Nimmo and W. Stothers (1994 / 2000), Fundamentals of University Mathematics, Albion Publishing / Horwood, ISBN 1898563101

H. Anton : Elementary Linear Algebra, Wiley (2000), ISBN 0471170550

October 10